## How Combinatorics became legitimate (according to László Lovász and Endre Szemerédi)

* Simons Foundation* has a series of fantastic interviews with leading mathematicians (ht Federico Ardila). Let me single out the interviews with László Lovász and Endre Szemerédi. Avi Wigderson asked both of them about the history of combinatorics and how it came into prominence. Watch parts 8-9 in Lovász’s interview and 10-11 in Szemerédi’s interview to hear their fascinating answers.

**P.S.** See also my old blog posts on what is combinatorics, how it became legitimate and how to watch math videos.

## Some good news

Two of my former Ph.D. students won major prizes recently — **Matjaž Konvalinka** and **Danny Nguyen**. Matjaž is an Associate Professor at University of Ljubljana, Danny is a Lewis Research Assistant Professor at University of Michigan, Ann Arbor. **Congratulations to both of them**!

(1) The * 2019 Robbins Prize *is awarded to

**Roger Behrend**,

**Ilse Fischer**and

**Matjaž Konvalinka**for their paper “

*Diagonally and antidiagonally symmetric alternating sign matrices of odd order*”. The

*Robbins Prize*is given in Combinatorics and related areas of interest is named after the late David P. Robbins and is given once every 3 years by AMS and MAA.

In many ways, this paper completes the long project of enumerating *alternating sign matrices* (ASMs) initiated by William Mills, David Robbins, and Howard Rumsey in the early 1980s. The original #ASM(*n*)=#TSSCPP(*n*) conjecture follows from Andrews’s proof of the conjectured product formula for #TSSCPP(*n*), and Zeilberger’s 84 page computer assisted proof of the the same conjectured product formula for #ASM(*n*). This led to a long series of remarkable developments which include Kuperberg’s proof using the *Izergin-Korepin determinant* for the six vertex model, the Cantini–Sportiello proof of the *Razumov-Stroganov conjecture,* and a recent self-contained determinantal proof for the number of ASMs by Fischer. Bressoud’s book (and this talk, slides) is a good introduction. But the full story is yet to be written.

(2) The ** 2018 Sacks Prize** is awarded to

**Danny Nguyen**for his UCLA Ph.D. dissertation on the complexity of short formulas in

*Presburger Arithmetic*(PA) and many related works (some joint with me, some with others). See also the UCLA announcement. The S

*acks Prize*is given by the international

**Association for Symbolic Logic**for “the most outstanding doctoral dissertation in mathematical logic“. It is sometimes shared between two awardees, and sometimes not given at all. This year Danny is the sole winner of the prize.

Danny’s dissertation is a compilation of eight (!) papers Danny wrote during his graduate studies, all on the same or closely related subject. These papers advance and mostly finish off the long program of understanding the boundary of what’s feasible in PA. The most important of these is our joint **FOCS paper** which basically says that *Integer Programming* and *Parametric Integer Programming* is all that’s left in **P**, while all longer formulas are **NP-hard**. See Featured MathSciNet Review by Sasha Barvinok and an overlapping blog post by Gil Kalai discussing these results. See also Danny’s FOCS talk video and my MSRI talk video presenting this work.

## Just combinatorics matters

I would really like everyone to know that every time you say or write that something is “just combinatorics” somebody rolls his eyes. Guess who?

Here is a short collection of “just combinatorics” quotes. It’s a followup on my “What is Combinatorics?” quotes page inspired by the “What is Combinatorics?” blog post.

## ICM Paper

Well, I finally finished my **ICM paper**. It’s only 30 pp, but it took many sleepless nights to write and maybe about 10 years to understand what exactly do I want to say. The published version will be a bit shorter – I had to cut section 4 to satisfy their page limitations.

Basically, I give a survey of various recent and not-so-recent results in *Enumerative Combinatorics* around three major questions:

**(1)** What is a formula?

**(2)** What is a good bijection?

**(3)** What is a combinatorial interpretation?

Not that I answer these questions, but rather explain how one *could answer* them from computational complexity point of view. I tried to cover as much ground as I could without overwhelming the reader. Clearly, I had to make a lot of choices, and a great deal of beautiful mathematics had to be omitted, sometimes in favor of the Computational Combinatorics approach. Also, much of the survey surely reflects my own POV on the subject. I sincerely apologize to everyone I slighted and who disagrees with my opinion! Hope you still enjoy the reading.

Let me mention that I will wait for a bit before posting the paper on the arXiv. I very much welcome all comments and suggestions! Post them here or email privately.

P.S. In thinking of how approach this paper, I read a large number of papers in previous ICM proceedings, e.g. papers by Noga Alon, Mireille Bousquet-Mélou, Paul Erdős, Philippe Flajolet, Marc Noy, János Pach, Richard Stanley, Benny Sudakov, and many others. They are all terrific and worth reading even if just to see how the field has been changing over the years. I also greatly benefited from a short introductory article by Doron Zeilberger, which I strongly recommend.

## Fibonacci times Euler

Recall the *Fibonacci numbers* given by 1,1,2,3,5,8,13,21… There is no need to define them. You all know. Now take the *Euler numbers (OEIS)* 1,1,1,2,5,16,61,272… This is the number of alternating permutations in with the exponential generating function . Both sequences are incredibly famous. Less known are connection between them.

(1) Define the *Fibonacci polytope* to be a convex hull of 0/1 points in with no two 1 in a row. Then has vertices and vol This is a nice exercise.

(2) (by just a little). For example, . This follows from the fact that

and , where is the g*olden ratio*. Thus, the product . Since and , the inequality is easy to see, but still a bit surprising that the numbers are so close.

Together with Greta Panova and Alejandro Morales we wrote a little note “Why is π < 2φ?” which gives a combinatorial proof of (2) via a direct surjection. Thus we obtain an indirect proof of the inequality in the title. The note is not a research article; rather, it is aimed at a general audience of college students. We will not be posting it on the arXiv, so I figure this blog is a good place to advertise it.

The note also explains that the inequality (2) also follows from Sidorenko’s theorem on complementary posets. Let me briefly mention a connection between (1) and (2) which is not mentioned in the note. I will assume you just spent 5 min and read the note at this point. Following Stanley, the volume of is equal to the volume of the *chain polytope* (=*stable set polytope*), see Two Poset Polytopes. But the latter is exactly the polytope that Bollobás, Brightwell and Sidorenko used in their proof of the upper bound via polar duality.

## You should watch combinatorics videos!

Here is my collection of links to *Combinatorics* *videos,* which I assembled over the years, and recently decided to publish. In the past few years the number of videos just exploded. We clearly live in a new era. This post is about how to handle the transition.

#### What is this new collection?

I selected over 400 videos of lectures and seminars in Combinatorics, which I thought might be of interest to a general audience. I tried to cover a large number of areas both within Combinatorics and related fields. I have seen many (but not all!) of the talks, and think highly of them. Sometimes I haven’t seen the video, but have heard this talk “live” at the same or a different venue, or read the paper, etc. I tried to be impartial in my selection, but I am sure there is some bias towards some of my favorite speakers.

The collection includes multiple lectures by Noga Alon, Persi Diaconis, Gil Kalai, Don Knuth, László Lovász, János Pach, Vic Reiner, Paul Seymour, Richard Stanley, Terry Tao, Xavier Viennot, Avi Wigderson, Doron Zeilberger, and many many others. Occasionally the speakers were filmed giving similar talks at different institutions, so I included quick links to those as well so the viewer can choose.

Typically, these videos are from some workshops or public lecture series. Most are hosted on the institution websites, but a few are on YouTube or Vimeo (some of these are broken into several parts). The earliest video is from 1992 and the most recent video was made a few days ago. Almost all videos are from the US or Canada, with a few recent additions from Europe. I also added links to a few introductory lectures and graduate courses on the bottom of the page.

#### Why now?

Until a couple of years ago, the videos were made only at a few conference centers such as Banff, MSRI and IAS. The choice was sparse and the videos were easy to find. The opposite is true now, on both counts. The number of recorded lectures in all areas is in tens of thousands, they are spread across the globe, and navigating is near impossible unless you know exactly what you are looking for. In fact, there are so many videos I really struggled with the choice of which to include (and also with which of them qualify as Combinatorics). I am not sure I can maintain the collection in the future – it’s already getting too big. Hopefully, some new technology will come along (see below), but for now this will do.

#### Why Combinatorics?

That’s what I do. I try to think of the area as broad as possible, and apologize in advance if I omitted a few things. For the subarea division, I used as a basis my own Wikipedia entry for Combinatorics (weirdly, you can listen to it now in a robotic voice). The content and the historical approach within sub-areas is motivated by my views here on what exactly is Combinatorics.

#### Why should you start watching videos now?

First, because you can. One of the best things about being in academia is the ability (in fact, necessity) to learn. You can’t possibly follow everything what happens in all fields of mathematics and even all areas of combinatorics. Many conferences are specialized and the same people tend to meet a year after year, with few opportunities for outsiders to learn what’s new over there. Well, now you can. Just scroll down the list and (hopefully) be amazed at the number of classical works (i.e. over 5 y.o.) you never heard of, the variety of recent developments and connections to other fields. So don’t just watch people in your area from workshops you missed for some reason. Explore other areas! You might be surprised to see some new ideas even on your favorite combinatorial objects. And if you like what you see, you can follow the links to see other videos from the same workshops, or search for more videos by the same speaker…

Second, you should start watching because it’s a very different experience. You already know this, of course. One can pause videos, go back and forward, save the video to watch it again, or stop watching it right in the beginning. This ability is to popular, Adam Sandler even made an awful movie about it… On the other hand, the traditional model of lecture attendance is where you either listen intently trying to understand in real time *and* take notes, or are bored out your mind but can’t really leave. It still has its advantages, but clearly is not always superior. Let me elaborate on this below.

#### How to watch videos?

This might seem like a silly question, but give me a chance to suggest a few ideas…

0) Prepare for the lecture. Make sure to have enough uninterrupted time. Lock the door. Turn off the cell phone. Download and save the video (see below). Download and save the slides. Search for them if they are not on the lecture website (some people put them on their home pages). Never delete anything – store the video on an external hard drive if you are running out of space. Trust me, you never know when you might need it again, and the space is cheap anyway…

Some years ago I made a mistake by not saving Gil Kalai’s video of a talk titled “Results and Problems around Borsuk’s Conjecture”. I found it very inspiring — it’s the only talk I referenced it in my book. Well, apparently, in its infinite wisdom, PIMS lost the video and now only the audio is available, which is nearly useless for a blackboard talk. What a shame!

1) Use 2 devices. Have the video on a big screen, say, a large laptop or a TV hooked to your laptop. If the TV is too far, use a wireless mouse to operate a laptop from across the room or something like a Google stick to project from a far. Then, have the slides of the talk opened on your tablet if you like taking computer notes or just like scrolling by hand gestures, or on your other laptop if you don’t. The slides are almost universally in .pdf and most software including the Adobe Reader allows to take notes straight in the file.

Another reason to have slides opened is the inability for some camera people to understand what needs to be filmed. This is especially severe if they just love to show the unusual academic personalities, or are used to filming humanities lectures where people read at the podium. As a result, occasionally, you see them pointing a camera to a slide full of formulas for 2 seconds (and out of focus), and then going back for 2 minutes filming a speaker who is animatedly pointing to the screen (now invisible), explaining the math. Ugh…

2) If the subject is familiar and you feel bored with the lengthy introduction, scroll the slides until you see something new. This will give you a hint to where you should go forward in the video. And if you did miss some definition you can pause the video and scroll the slides to read it.

3) If there are no slides, or you want to know some details which the speaker is purposefully omitting, pause the video and download the paper. I do this routinely while listening to talks, but many people are too shy to do this out of misplaced fear that others might think they are not paying attention. Well, there is no one to judge you now.

4) If you are the kind of person who likes to ask questions to clarify things, you still can. Pause the video and search the web for the answer. If you don’t find it, ask a colleague by skype, sms, chat, email, whatever. If everything fails – write to the speaker. She or he might just tell you, but don’t be surprised if they also ignore your email…

5) If you know others who might be interested in the video lecture, just make it happen. For example, you can organize a weekly seminar where you and your graduate students watch the lectures you choose (when you have no other speakers). If students have questions, pause the video and try to answer them. In principle, if you have a good audience the speaker may agree to answer the questions for 5-10 min over skype, after you are done watching. Obviously, I’ve never seen this happen (too much coordination?). But why not try this – I bet if you ask nicely many speakers would agree to this.

6) If you already know a lot about the subject, haven’t been following it recently but want to get an update, consider binge watching. Pick the most recent lecture series and just let it run when you do house shores or ride a subway. When things get interesting, you will know to drop everything and start paying attention.

#### Why should you agree to be videotaped?

Because the audience is ready to see your talks now. Think of this as another way of reaching out with your math to a suddenly much broader mathematical community (remember the “broad impact” section on your NSF grant proposal?). Let me just say that there is nothing to fear – nobody is expecting you to have acting skills, or cares that you have a terrible haircut. But if you make a little effort towards giving a good talk, your math will get across and you might make new friends.

Personally, I am extremely uncomfortable being videotaped – the mere knowledge of the camera filming makes me very nervous. However I gradually (and grudgingly) concluded that this is now a part of the job, and I have to learn how to do this well. Unfortunately, I am not there yet…

Yes, I realize that many traditionalists will object that “something will be missing” when you start aiming at giving good video talks at the expense of local audience. But the world is changing if hasn’t changed already and you can’t stop the tide. This happened before, many times. For example, at some point all the big Hollywood studios have discovered that they can make movies simpler and make a great deal more money overseas to compensate for the loss in the US market. They are completely hooked now, and no matter what critics say this global strategy is likely irreversible. Of course, this leaves a room for a niche market (say, low budget art-house movies), but let’s not continue with this analogy.

#### How to give video lectures?

Most people do nothing special. Just business as usual, hook up the mike and hope it doesn’t distort your voice too bad. That’s a mistake. Let me give a number of suggestions based mostly on watching many bad talks. Of course, the advice for giving regular talks apply here as well.

0) Find out ahead of time if you get filmed and where the camera is. During the lecture, don’t run around; try to stand still in full view of the camera and point to the screen with your hands. Be animated, but without sudden moves. Don’t use a laser pointer. Don’t suddenly raise your voice. Don’t appeal to the previous talks at the same workshop. Don’t appeal to people in the audience – the camera can rarely capture what they say or do. If you are asked a question, quickly summarize it so the viewer knows what question you are answering. Don’t make silly off-the-cuff jokes (this is a hard one).

1) Think carefully whether you want to give a blackboard or a computer talk. This is crucial. If it’s a blackboard talk, make sure your handwriting is clear and most importantly BIG. The cameras are usually in the very back and your handwriting won’t be legible otherwise. Unless you are speaking the Fields Institute whose technology allows one to zoom into the high resolution video, nobody might be able to see what you write. Same goes for handwritten slides unless they are very neat, done on a laptop, and the program allows you to increase their size. Also, the blackboard management becomes a difficult issue. You should think through what results/definitions should stay on the blackboard visible to the camera at all times and what can be safely deleted or lifted up if the blackboard allows that.

2) If it’s a computer talk, stick to your decision and make a lot of effort to have the slides look good. Remember, people will be downloading them… Also, make every effort NOT to answer questions on a blackboard next to the screen. The lightning never works – the rooms are usually dimmed for a computer talk and no one ever thinks of turning the lights on just for 30 seconds when you explain something. So make sure to include all your definition, examples, etc, in the slides. If you don’t want to show some of them – in PowerPoint there is a way to hide them and pull them up only if someone asks to clarify something. I often prepare the answers to some standard questions in the invisible part of my slides (such as “What happens for other root systems?” or “Do your results generalize to higher dimensions?”), sometimes to unintended comedic effect. Anyhow, think this through.

3) Don’t give the same videotaped talk twice. If you do give two or more talks on the same paper, make some substantial changes. Take Rota’s advice: “Relate to your audience”… If it’s a colloquium talk, make a broad accessible survey and include your results at the end. Or, if it’s a workshop talk, try to make an effort to explain most proof ideas, etc. Make sure to have long self-explanatory talk titles to indicate which talk is which. Follow the book industry lead for creating subtitles. For example use “My most recent solution of the Riemann hypothesis, an introduction for graduate students” or “The Pythagorean theorem: How to apply it to the Langlands Program and Quantum Field Theory”.

4) Download and host your own videos on your website alongside your slides and your relevant paper(s), or at least make clear links to them from your website. You can’s trust anyone to keep your files. Some would argue that re-posting them on YouTube will then suffice. There are two issues here. First, this is rarely legal (see below). Second, as I mentioned above, many viewers would want to have a copy of the file. Hopefully, in the future there will be a copyright-free arXiv-style video hosting site for academics (see my predictions below).

5) In the future, we would probably need to consider having a general rule about adding a file with errata and clarifications to your talk, especially if something you said is not exactly correct, or even just to indicate, post-factum, whether all these conjectures you mentioned have been resolved and which way. The viewers would want to know.

For example, my student pointed out to me that in my recent Banff talk, one of my lemmas is imprecise. Since the paper is already available, this is not a problem, but if it wasn’t this could lead to a serious confusion.

6) Watch other people’s videos. Pay attention to what they do best. Then watch your own videos. I know, it’s painful. Turn off the sound perhaps. Still, this might help you to correct the worst errors.

7) For advanced lecturers – try to play with the format. Of course, the videos allow you to do things you couldn’t do before (like embedding links to papers and other talks, inserting some Java demonstration clips, etc.), but I am talking about something different. You can turn the lecture into an artistic performance, like this amazing lecture by Xavier Viennot. Not everyone has the ability or can afford to do this, but having it recorded can make it worthwhile, perhaps.

#### Know your rights

There are some delicate legal issues when dealing with videos, with laws varying in different states in the US (and in other countries, of course). I am not an expert on any of this and will write only as I understand them in the US. Please add a comment on this post if you think I got any of this wrong.

1) Some YouTube videos of math lectures look like they have been shut by a phone. I usually don’t link to those. As I understand the law on this, anyone can film a public event for his/her own consumption. However, you and the institution own the copyright so the YouTube posting is illegal without both of yours explicit permission (written and signed). You can fight this by sending a “cease and desist” letter to the person who posted the video, but contacting YouTube directly might be more efficient – they have a large legal department to sort these issues.

2) You are typically asked to sign away your rights before your talk. If an institution forgot to do this, you can ask to take your talk down for whatever reason. However, even if you did sign the paper you can still do this – I doubt the institution will fight you on this just to avoid bad publicity. A single email to the IT department should suffice.

3) If the file with your talk is posted, it is (obviously) legal for you to download it, but not to post it on your website or repost elsewhere such as YouTube or WordPress. As far as I am concerned, you should go ahead and post it on your university website anyway (but not on YT or WP!). Many authors typically post all their papers on their website even if they don’t own a copyright on them (which is the case or virtually all papers before 2000). I am one of them. The publishers just concluded that this is the cost of doing business – if they start going after people like us, the authors can revolt. The same with math videos. The institutions probably won’t have a problem with your university website posting as long as you acknowledge the source. But involving a third party creates a host of legal problems since these internet companies are making money out of the videos they don’t own a copyright for. Stay away from this.

4) You can the edit the video by using numerous software, some of which is free to download. Your can remove the outside noise, make the slides sharper, everything brighter, etc. I wouldn’t post a heavily edited video when someone else owns a copyright, but a minor editing as above is ok I think.

5) If the institution’s website does not allow to download the video but has a streaming option (typically, the Adobe Flash or HTML5), you can still legally save it on your computer, but this depends on what software you choose. There are plenty of software which capture the video being played on your computer and save it in a file. These are 100% legal. Other websites play the videos on *their* computers and allow you to download afterwards. This is probably legal at the institutions, but a gray area at YouTube or Vimeo which have terms of service these companies may be violating. Just remember – such videos can only be legal for personal consumption. Also, the quality of such recording is typically poor – having the original file is much better.

#### What will happen in the future?

Yes, I will be making some predictions. Not anything interesting like Gian-Carlo Rota’s effort I recently analyzed, but still…

1) Watching and giving video lectures will become a norm for everyone. The ethical standards will develop that everyone gets to have the files of videos they made. Soon enough there will be established some large well organized searchable (and not-for-profit!) math video depositories arXiv-style where you can submit your video and link to it from your website and where others can download from. Right now companies like DropBox allow you to do this, but it’s for-profit (your have to pay extra for space), and it obviously needs a front like the arXiv. This would quickly make my collection a thing of the past.

2) Good math videos will become a “work product”, just like papers and books. It is just another venue to communicate your results and ideas. People will start working harder on them. They will become a standard item on CVs, grant applications, job promotions, etc. More and more people will start referencing them just like I’ve done with Kalai’s talk. Hopefully part 1) will happen soon enough so all talks get standard and stable links.

3) The video services will become ubiquitous. First, all conference centers will acquire advanced equipment in the style of the Banff Center which is voice directed and requires no professional involvement except perhaps at the editing stage. Yes, I am thinking of you, MFO. A new library is great, but the talks you could have recorded there are priceless – it’s time to embrace the 21st century….

Second, more and more university rooms will be equipped with the cameras, etc. UCLA already has a few large rooms like that (which is how we make the lamely named BruinCasts), but in time many department will have several such rooms to hold seminars. The storage space is not an issue, but the labor cost, equipment and the broadband are. Still, I give it a decade or two…

4) Watching and showing math videos will become a standard part of the research and graduate education. Ignore the doomsayers who proclaim that this will supplant the traditional teaching (hopefully, not in our lifetime), but it’s clear already there are unexplored educational benefits from this. This should be of great benefit especially to people in remote locations who don’t have access to such lectures otherwise. Just like the Wikipedia has done before, this will even the playing field and help the talent to emerge from unlikely places. If all goes well, maybe the mathematics will survive after all…

**Happy watching everyone! **

## Grading Gian-Carlo Rota’s predictions

In this post I will try to evaluate Gian-Carlo Rota‘s predictions on the future of Combinatorics that he made in this 1969 article. He did surprisingly well, but I am a tough grader and possibly biased about some of the predictions. Judge for yourself…

#### It’s tough to make predictions, especially about the future

It is a truth universally acknowledged that humans are very interested in predicting the future. They do this incessantly, compiling the lists of the best and the worst, and in general can’t get enough of them. People tend to forget wrong predictions (unless they are outrageously wrong). This allows a person to make the same improbable predictions over and over and over and over again, making news every time. There are even professional prognosticators who make a living writing about the future of life and technology. Sometimes these predictions are rather interesting (see here and there), but even the best ones are more often wrong than right…

Although rarely done, analyzing past predictions is a useful exercise, for example as a clue to the truthiness of the modern day oracles. Of course, one can argue that many of the political or technology predictions linked above are either random or self-serving, and thus not worth careful investigation. On the other hand, as we will see below, Rota’s predictions are remarkably earnest and sometimes even brave. And the fact that they were made so long ago makes them uniquely attractive, practically begging to be studied.

Now, it has been 45 years since Rota’s article, basically an eternity in the life span of Combinatorics. There, Rota describes Combinatorics as “*the least developed branches of mathematics*“. A quick review of the last few quotes on this list I assembled shows how much things have changed. Basically, the area moved from an ad hoc collection of problems to a 360-degree panorama of rapidly growing subareas, each of which with its own deep theoretical results, classical benchmarks, advanced tools and exciting open problems. This makes grading rather difficult, as it suggests that even random old predictions are likely to be true – just about anything people worked on back in the 1960 has been advanced by now. Thus, before turning to Rota, let’s agree on the grading scale.

#### Grading on a curve

To give you the feel for my curve, I will use the celebrated example of the 1899-1901 postcards in the En L’An 2000 French series, which range from insightful to utter nonsense (click on the titles to view the postcards, all available from Wikimedia).

**•** * Electric train*. Absolutely. These were introduced only in 1940s and have been further developed in France among other countries. Note the aerodynamic shape of the engine. Grade:

**A**.

**•** * Correspondance cinema. *Both the (silent) cinema and phonograph were invented by 1900; the sound came to movie theaters only in 1927. So the invention here is of a home theater for movies with sound. Great prediction although not overly ambitious. Grade:

**A-**.

**•** ** Military cyclists**. While bicycle infantry was already introduced in France by 1900, military use of motorcycles came much later. The idea is natural, but some designs of bikes from WW2 are remarkably similar. Some points are lost due to the lack of widespread popularity in 2000. Grade:

**B+**.

**•** ** Electric scrubbing**. This is an electric appliance for floor cleaning. Well, they do exist, sort of, obviously based on different principles. In part due to the modern day popularity, this is solid prediction anyway. Grade:

**B**.

**•** * Auto-rollers*. Roller skates have been invented in 18th century and by 1900 became popular. So no credit for the design, but extra credit for believing in the future of the mean of transportation now dominated by rollerblades. Thus the author’s invention is in the category of “motorized personal footwear”. In that case the corresponding modern invention is of the electric skateboard which has only recently become available, post-2000 and yet to become very popular. Grade:

**B-**.

**•** ** Barber**. The author imagines a barber operating machinery which shaves and cuts customer’s hair. The design is so ridiculous (and awfully dangerous), it’s a good thing this never came about. There are however electric shavers and hair cutters which are designed very differently. Grade:

**C**.

* • Air cup*. The Wright brothers’ planes had similar designs, so no credit again. The author assumes that personal air travel will become commonplace, and at low speeds and heights. This is almost completely false. However, unfortunately, and hopefully only very occasionally, some pilots do enjoy one for the road. Grade:

**D**.

**•** * Race in Pacific*. The author imagines that the public spectacle of horse racing will move underwater and become some kind of fish racing. Ridiculous. Also a complete failure to envision modern popularity of auto racing which already began in Paris in 1887. Grade:

**F**.

#### Rota’s predictions on combinatorial problems:

In his paper, Rota writes:

Fortunately, most combinatorial problems can be stated in everyday language. To give an idea of the present state of the field, we have selected a few of the many problems that are now being actively worked upon.

We take each of these “problems” as a kind of predictions of where the field is going. Here are my (biased and possibly uninformed) grades for each problem he mentions.

**1) ** **The Ising Problem****.** I think it is fair to say that since 1969 combinatorics made no contribution in this direction. While physicists and probabilists continue studying this problem, there is no exact solution in dimension 3 and higher. Grade: **F**.

**2) ** ** Percolation Theory.** The study of percolation completely exploded since 1969 and is now a subject of numerous articles in both probability, statistical physics and combinatorics, as well as several research monographs. One connection is given by an observation that

*p*-percolation on a complete graph

*K*gives the Erdős–Rényi random graph model. Even I accidentally wrote a few papers on the subject some years ago (see one, two and three). Grade:

_{n}**A**.

**3)** * The Number of Necklaces, and Polya’s Problem.* Taken literally, the necklaces do come up in combinatorics of words and free Lie algebra, but this context was mentioned by Rota already. As far as I can tell, there are various natural (and interesting) generalizations of necklaces, but none surprising. Of course, if the cyclic/dihedral group action here is replaced by other actions, e.g. the

*symmetric group*, then modern developments are abundant. But I think it’s a reach too far, since Rota knew the works of Young, MacMahon, Schur and others but does not mention any of it. Similarly, Polya’s theorem used to be included in all major combinatorics textbooks (and is included now, occasionally), but is rarely taught these days. Simply put, the g.f. implications haven’t proved useful. Grade:

**D**.

**4)** * Self-avoiding Walk.* Despite strong interest, until recently there were very few results in the two-dimensional case (some remarkable results were obtained in higher dimensions). While the recent breakthrough results (see here and there) do use some interesting combinatorics, the authors’ motivation comes from probability. Combinatorialists did of course contribute to the study of somewhat related questions on enumeration of various classes of polyomino (which can be viewed as self-avoiding cycles in the grid, see e.g. here). Grade:

**C**.

**5)** ** The Traveling Salesman Problem.** This is a fundamental problem in optimization theory, connected to the study of Hamiltonian cycles in Graph Theory and numerous other areas. It is also one of the earliest NP-hard problems still playing a benchmark role in Theoretical Computer Science. No quick of summary of the progress in the past 45 years would give it justice. Note that Rota’s paper was written before the notions of NP-completeness. In this light, his emphasis on algorithmic complexity and allusions to Computability Theory (e.g. unsolvable problems) are most prescient. So are his briefly mentioned connections to topology which are currently a popular topic. Well done! Grade:

**A+**.

**6)** * The Coloring Problem.* This was a popular topic way before Rota’s article (inspired by the Four Color Theorem, the chromatic polynomial, etc.), and continues to be even more so, with truly remarkable advances in multiple directions. Note Rota’s mentioning of matroids which may seem extraneous here at first, but in fact absolutely relevant indeed (in part due to Rota’s then-ongoing effort). Very good but unsurprising prediction. Grade:

**A-**.

**7) ** * The Pigeonhole Principle and Ramsey’s Theorem.* The Extremal Graph Theory was about to explode in many directions, with the the Erdős-Stone-Simonovits theorem proved just a few years earlier and the Szemerédi regularity lemma a few years later. Still, Rota never mentions Paul Erdős and his collaborators, nor any of these results, nor potential directions. What a missed opportunity! Grade:

**B+**.

#### Rota’s predictions on combinatorial areas:

In the concluding section “The Coming Explosion”, Rota sets this up as follows:

Before concluding this brief survey, we shall list the main subjects in which current work in combinatorial theory is being done.

Here is a list and more of my comments.

**1)** ** Enumerative Analysis.** Sure. But this was an easy prediction to make given the ongoing effort by Carlitz, Polya, Riordan, Rota himself and many other peope. Grade:

**A-**.

**2)** * Finite Geometries and Block Designs.* The subject was already popular and it did continue to develop but perhaps at a different pace and directions than Rota anticipated (Hadamard matrices, tools from Number Theory). In fact, a lot of later work was connection with with Group Theory (including some applications of CFSG which was an ongoing project) and in Coding Theory (as Rota predicted). Grade:

**B-**.

**3)** ** Applications to Logic.** Rota gives a one-sentence desctiption:

The development of decision theory has forced logicians to make wide use of combinatorial methods.

There are various important connections between Logic and Combinatorics, for example in Descriptive Set Theory (see e.g. here or more recent work by my future UCLA colleague there). Note however, that Infinitary Combinatorics was already under development, after the Erdős-Rado theorem (1956). Another very interesting and more recent connection is to Model Theory (see e.g. here). But the best interpretation here I can think of here are the numerous applications to Game Theory, which already existed (Nash’s equilibrium theorem is from 1950) and was under rapid development. Either way, Rota was too vague in this case to be given much credit. Grade: **C**.

**4)** ** Statistical Mechanics.** He mentions the Ising model again and insists on “close connections with number theory”. One can argue this all to be too vague or misdirected, but the area does indeed explode in part in the directions of problems Rota mentions earlier. So I am inclined to give him benefit of the doubt on this one. Grade:

**A-**.

#### The final grade

In total, Rota clearly got more things right than wrong. He displayed an occasional clairvoyance, had some very clever insights into the future, but also a few flops. Note also the near complete lack of self-serving predictions, such as the Umbral Calculus that Rota was very fond of. Since predictions are hard, *successes* have a great weight than failures. I would give a *final grade* somewhere between **A-** and **B+** depending on how far into the future do we think he was making the predictions. Overall, good job, Gian-Carlo!

**P.S.** Full disclosure: I took a few advanced classes with Gian-Carlo Rota as a graduate student cross registering from Harvard to MIT, and he may have graded my homeworks (this was in 1994-1996 academic years). I don’t recall the final grades, but I think they were good. Eventually Rota wrote me a letter of recommendation for a postdoc position.